Since their first demonstration, modelocked thin-disk lasers have consistently surpassed other modelocked oscillator technologies in terms of achievable pulse energy and average power by several orders of magnitude. Surprisingly, state-of-the-art results using this technology have so far only been achieved in modelocking regimes where soliton pulse shaping is dominant (i.e., soliton modelocking with semiconductor saturable absorber mirrors or Kerr lens modelocking), in which only small nonlinear phase shifts are tolerable, ultimately limiting pulse energy scaling. Inspired by the staggering success of novel modelocking regimes applied to overcome these limitations in modelocked fiber lasers, namely the similariton (self-similarly evolving pulses) and dissipative soliton regimes, here, we explore these nonlinearity-resistant regimes for the next generation of high-energy modelocked thin-disk lasers, whereby millijoule pulse energies appear to be realistic targets. In this goal, we propose two possible implementations. The first is based on a passive multipass cell and designed to support dissipative solitons in an all-normal dispersion cavity. The second incorporates an active multipass cell and is designed to support similaritons. Our numerical investigations indicate that this is a very promising path to increase the pulse energy achievable directly from modelocked oscillators toward the millijoule level while additionally simplifying their implementation by eliminating the need for operation in cumbersome vacuum chambers.